Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7519 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#7520 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1402 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | #492 | C23 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4)(5,11,14,10)(6,15,16,9)(7,12)(8,13,18,17);(3,5,7,14)(4,8,12,15)(6,16,17,13)(9,10,18,11)44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#153628, C2×#153642, C2×#153472, … |
| maximal normal subgroup of | C2×#153628, C2×#153642, C2×#153472, … |
| maximal quotient of | #72283634 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60642);